It is demonstrated that, making minimal changes in ordinary quantum mechanics, a reasonable irreversible quantum mechanics can be obtained. This theory has a more general spectral decomposition, with eigenvectors corresponding to unstable states that vanish when $t\to \infty .$ These Gamov vectors have zero norm, in such a way that the norm and the energy of the physical states remain constant. The evolution operator has no inverse, showing that we are really dealing with a time-asymmetric theory. Using the Friedrichs model, reasonable physical results are obtained, e.g., the remnant of an unstable decaying state reappears, in the continuous spectrum of the model, with its primitive energy.

A method is proposed for the efficient calculation of the Green’s functions and eigenstates for quantum systems of two or more dimensions. For a given Hamiltonian, the best possible separable approximation is obtained from the set of all Hilbert-space operators. It is shown that this determination itself, as well as the solution of the resultant approximation, is a problem of reduced dimensionality. Moreover, the approximate eigenstates constitute the optimal separable basis, in the sense of self-consistent field theory. The full solution is obtained from the approximation via iterative expansion. In the time-independent perturbation expansion for instance, all of the first-order energy corrections are zero. In the Green’s function case, we have a distorted-wave Born series with optimized convergence properties. This series may converge even when the usual Born series diverges. Analytical results are presented for an application of the method to the two-dimensional shifted harmonic-oscillator system, in the course of which the quantum ${\tanh }^2$ potential problem is solved exactly. The universal presence of bound states in the latter is shown to imply long-lived resonances in the former. In a comparison with other theoretical methods, we find that the reaction path Hamiltonian fails to predict such resonances.

This paper extends previous results about the classical information capacity of a noiseless quantum-mechanical communication channel to situations in which the final signal states are mixed states, that is, to channels with noise.

The spinless Salpeter equation may be considered either as a standard approximation to the Bethe-Salpeter formalism, designed for the description of bound states within a relativistic quantum field theory, or as the most simple, to a certain extent relativistic, generalization of the costumary nonrelativistic Schrödinger formalism. Because of the presence of the rather difficult-to-handle square-root operator of the relativistic kinetic energy in the corresponding Hamiltonian, very frequently the corresponding (discrete) spectrum of energy eigenvalues cannot be determined analytically. Therefore, we show how to calculate, by some clever choices of basis vectors in the Hilbert space of solutions, for the rather large class of power-law potentials, at least upper bounds on these energy eigenvalues. For the lowest-lying levels, this may be done even analytically.

We show that there is essentially only one way to construct a stochastic Schrödinger equation that gives a dynamical account of the transformation of entangled into factorized states and is consistent with both quantum mechanics and required symmetries. The noisy, nonlinear term is a unimodular scalar multiple of the time-reversal operator that must be present whenever a Hamiltonian term in the Schrödinger equation can distinguish the factorized constituents of an entangled state. The dynamical mechanism involved in the transformation of entangled into factorized states provides an explanation for the fact that Einstein-Podolsky-Rosen correlations appear in a time determined by the response of the measuring device and independent of the distance between the particles. The dependence on the response time of the measuring device may be testable through a delay in observing the collapse of mesoscopic “Schrödinger cat” states in ion traps. It is further shown that there are situations where a two-particle interaction can induce a nonlinear term by virtue of coupling to decay modes that distinguish factorized constituents of an entangled state. We show that this should happen in the neutral $K$-meson system where the entangled $K_L$ state is pushed slightly in the direction of a factorized constituent ${(K}_0$ or $\overline{K_0})$ as a consequence of the fact that these can be distinguished via the sign of the charged lepton in a semileptonic decay mode. The result is a $\mathrm{CP}$ violation that is within 20% of the experimental value.

The Pancharatnam phase in two-photon experiments is interpreted in terms of a pair of paths in the product space of polarization states. Vector-length holonomy in Weyl space is identified with the transition amplitude for a two-level orthogonally polarized light system. It is shown that the geometry of the Poincaré sphere admits Weyl structure.

Quantum-classical mixing is studied by a group-theoretical approach, and a quantum-classical equation of motion is derived. The quantum-classical bracket entering the equation preserves the Lie algebra structure of quantum and classical mechanics, and, therefore, leads to a natural description of interaction between quantum and classical degrees of freedom. The exact formalism is applied to coupled quantum and classical oscillators. Various approximations, such as the mean-field and the multiconfiguration mean-field approaches, which are of great utility in studying realistic multidimensional systems, are derived. Based on the formulation, a natural classification of the previously suggested quantum-classical equations of motion arises, and several problems from earlier works are resolved.

We report the results of an experiment in which postselected polarization-entangled photons were used to test the recent nonlocality proof by L. Hardy [Phys. Rev. Lett. 71, 1665 (1993)]. The particular source of entangled photons used in the experiment has the advantage that it is very easy to prepare a pair of photons in a state with any degree of entanglement. Hardy’s theorem, which only works for nonmaximally entangled states, was tested for a range of states.

We present a semiclassical description of the level density of a two-dimensional circular quantum dot in a homogeneous magnetic field. We model the total potential (including electron-electron interaction) of the dot containing many electrons by a circular billiard, i.e., a hard-wall potential. Using the extended approach of the Gutzwiller theory developed by Creagh and Littlejohn, we derive an analytic semiclassical trace formula. For its numerical evaluation we use a generalization of the common Gaussian smoothing technique. In strong fields orbit bifurcations, boundary effects (grazing orbits) and diffractive effects (creeping orbits) come into play, and the comparison with the exact quantum-mechanical result shows major deviations. We show that the dominant corrections stem from grazing orbits, the other effects being much less important. We implement the boundary effects, replacing the Maslov index by a quantum-mechanical reflection phase, and obtain a good agreement between the semiclassical and the quantum result for all field strengths. With this description, we are able to explain the main features of the gross-shell structure in terms of just one or two classical periodic orbits.

We discuss a simple method to probe the wave function $\psi \mathrm{}\left(x\right)\mathrm{}$ describing the transverse center-of-mass motion of an atomic beam. We detect position distributions of the atoms after they have interacted with two counterpropagating laser beams. The resulting interference pattern depends crucially on the cross term ${\psi }^{*}\mathrm{}\left(x\right)\mathrm{}\psi \left(x-\Delta x\right)$, which finally allows us to reconstruct the phase of $\psi \mathrm{}\left(x\right)\mathrm{}$.

We consider the Stark effect for the ground state of hydrogen. Using the Borel summability of the Rayleigh-Schrödinger perturbation expansion we sum it about the pure imaginary field strength $\mathcal{E}=i\mathcal{F},$ F is real. As a result we obtain the series converging for small values of F. We give arguments that this series converges for all positive finite values of F. The series obtained can be continued analytically back to the real field strength region. This method allows one to obtain accurate results for the hydrogenic Stark resonances with slight computational effort.

We show that the Calogero and Calogero-Sutherland models possess an $N$-body generalization of shape invariance. We obtain the operator representation that gives rise to this result, and discuss the implications of this result, including the possibility of solving these models using algebraic methods based on this shape invariance. Our representation gives us a natural way to construct supersymmetric generalizations of these models, which are interesting both in their own right and for the insights they offer in connection with the exact solubility of these models.

We analyze the motion of a Rydberg electron in combined external electric and magnetic fields using different levels of classical perturbation theory. Important dynamical properties of these systems are captured analytically through adiabatic invariants, and three widely different time scales for motion are identified. The hydrogen atom in tilted electric and magnetic fields, the hydrogen atom in combined magnetic and circularly polarized microwave fields, and the generalized van der Waals interaction are among the cases we examine.

A model of the time evolution of two interacting Rydberg manifolds of energy levels subject to a linearly ramped electric field is solved exactly in the Landau-Zener (LZ) approximation. Each manifold’s levels are treated as linear in time, parallel, equally spaced, and infinite in number. Their pairwise interactions produce a regular two-dimensional grid of isolated anticrossings. The time development of an initially populated state is then governed by two-level LZ transitions at avoided crossings and adiabatic evolution between them, parametrized by the LZ transition probability $D$ and a dynamical phase unit φ. The resulting probability distributions of levels are given analytically in the form of recursion relations, generating functions, integral representations involving $D$ and φ, and in certain limits by Bessel or Whittaker functions. Level populations are mapped out versus location on the grid for a range of cases. Interference effects lead to two principal types of probability distributions: a braiding adiabatic pattern with revivals for small $D$ and a diabatic pattern for $D\to 1$ in which only certain levels parallel to the initial one are appreciably populated. The sensitivity of the coherent evolution to φ is discussed, along with the relation of this model to others and to selective-field ionization.

The lifetime of an unstable atom can be extended by watching it closely, i.e., illuminating it with an intense electromagnetic field of appropriate frequency. This is an example of “dominated evolution” and is closely related to the so-called “quantum Zeno effect.” For a metastable atom bathed in a laser beam at the frequency of another of its transitions, we obtain an expression for the modified lifetime as a function of beam intensity. This provides an example of the quantum Zeno effect on a truly decaying system, and also should be useful for probing short distance features of atomic wave functions.

The ground-state hyperfine splitting values of high-$Z$ hydrogenlike ions are calculated. The relativistic, nuclear, and QED corrections are taken into account. The nuclear magnetization distribution correction (the Bohr-Weisskopf effect) is evaluated within the single-particle model with the $g_S$ factor chosen to yield the observed nuclear moment. An additional contribution caused by the nuclear-spin-orbit interaction is included in the calculation of the Bohr-Weisskopf effect. It is found that the theoretical value of the wavelength of the transition between the hyperfine-splitting components in ${}^{165}{\mathrm{Ho}}^{66+}$ is in good agreement with experiment.

Extreme narrowing, or collapse, of electron-vibrational bands is predicted as a new phenomenon in resonant radiative and nonradiative x-ray scattering (RXS) spectra. It is shown that in the inelastic scattering case, that is, when the potential surfaces of ground and final states are different, a considerable narrowing of the vibronic RXS band results from detuning the excitation photon frequency $\omega$ away from the absorption resonance. By fine tuning the frequency, this may under special circumstances also occur in the region of strong photoabsorption. In the case of elastic Rayleigh scattering, that is, when the potential surfaces of ground and final states are identical, the narrowing results in a total collapse to a single resonance by detuning the frequency. The theory predicts how the differencies in vibrational structure of resonant and nonresonant photoemission spectra depends on the excitation frequency. The effect of frequency detuning on complex spectral multilevel structures due to multiplet and spin-orbit splittings is discussed qualitatively. It is shown that the notion of duration time for the x-ray scattering process plays a crucial role in the understanding of RXS spectra.

Hyperfine separations in unstable ${\mathrm{Eu}}^{+}$ ions of mass 148, 149, and 150 have been measured in laser-microwave double-resonance experiments in a Paul ion trap. In spite of the small available quantities of the isotopes, the experimental uncertainties are of the order of ${10}^{-8}$ or below, which is of the same order as in earlier measurements on stable isotopes of ${\mathrm{Eu}}^{+}.$ Extensive second-order perturbation calculation is required to obtain coupling constants for magnetic-dipole $\mathrm{}\left(A\right)\mathrm{}$ and electric-quadrupole $\mathrm{}\left(B\right)\mathrm{}$ interactions. The uncertainties are a few times ${10}^{-7}$ for $A$ and ${10}^{-3}$ for $B.\mathrm{}$ The experiments are a step in an attempt to determine the differential hyperfine anomaly (Bohr-Weisskopf effect) in a long chain of isotopes.

Three-body corrections of order ${\alpha }^6{m}^2{c}^2/M$ to the fine-structure splittings of helium and three-body terms of order ${\alpha }^6{\mathrm{mc}}^2$ in lithium are derived and expressed in terms of expectation values of nonrelativistic operators. Corrections of order ${\alpha }^6\mu c^2$ to non-$S$ levels in two-body systems are also calculated. Perturbative calculation of one-body kinetic energy up to order ${10}^6$ of powers of $\alpha$ agrees with nonperturbatively numerical calculation within 19 significant figures. Such agreement provides a direct demonstration of the correctness of perturbation theory and of a simple technique developed previously [T. Zhang, Phys. Rev. A 54, 1252 (1996)] to calculate corrections arising from severely divergent nonrelativistic operators derived from the bound-state QED and relativistic kernels.